Friday, March 25, 2011

Did you know...?

Did you know that math is all made up?

This post is inspired by a tweet from David Cox referring to a project one of his students must've been presenting:!/dcox21/status/50600806541561856

As a math teacher, I get a lot of students who have learned math algorithmically. Even my "best" students in calculus often come in knowing how to do the work, but can't go much deeper than that. I think that's because math is often taught that way.

Sometimes I feel that way, too. I start to think, "If they can't understand the deeper meanings behind the subject, the least I can get them to do is follow a set of instructions to pass the high-stakes test at the end of the year." The problem is, many students can't even do that. So, I get frustrated that I laid out the 5 steps to factoring a trinomial and they still can't do it.

The point is, students see math as something coming down from the mountain on stone tablets. Reasons for either loving or hating math include: "There is only one correct answer to every problem" and "All you have to do is follow the steps and it works out."

Ahhh! Math!

Almost never have I been asked "Why?" in my math classes (other than "why are we doing this?").

As a kid I was the same way, but I was that way about most things. It never occurred to me to ask why the painter put a tree here and not there in his landscape; that must've been just the way it looked. I never thought to ask why the author used this word choice over another because that's just how you say something. Similarly in math class, I never thought to ask why our numbers look the way they do (why does the symbol 10 come after the symbol 9?) or why positives are on the right of the number line or why y's go over x's for slope. I was a good kid. I followed the rules. Someone older (and thus much wiser, of course) told me that's how it was and I took that to mean That's How It Is(TM).

But, did you know that math is completely man-made in the same way as the painting or novel? One plus one means absolutely nothing. Someone could've made a system where one plus one equals five and made up any rules they wanted. Logically, there's no real reason that the positives are on the right and up and negatives are on the left and down on the xy-plane. In fact, that's an arbitrary choice that the rest of the math doesn't even care about.

This was a great revelation to me when I finally "got it" in college (as a math major). We make it up.

Now, like any good system that sticks around, there has to be something in it that resonates with something else. Why have we all seen the Mona Lisa, but only a few people have seen your finger-painting from when you were 4? Because there's something common and interesting about Mona Lisa that resonates with a large audience. Your finger painting only meant something to you and your family, so they are the only ones who kept it in memory. Similar with these math choices (or my understanding of how evolution works). You can try anything you want, but only some things will stick because they are interesting or useful enough to propagate the species.

Bodies are an adult construct

Sure, the positive part of the x-axis could go to the left. But why? Since it's an arbitrary choice, why not go with our psychological convention that positive goes right. Then, when we need a new axis to represent a distinct direction, it seems to make the most sense to draw it vertically and have the positive part point up. Visually and even emotionally it feels right to do that.

So, why do we have the slope formula the way we do? Well, it makes sense to me that things like y = x that "go up" when you read from left to right should have a positive slope. Psychologically a graph that "increases" is a "good thing" and thus "positive." Now, how should we compare the graphs of y = x versus the graph of y = 2x? Well, the second one goes up faster, so it makes sense to me that it should be "more positive" so the slope should be a bigger number. With delta-y/delta-x you get 1 for the first and 2 for the second. With delta-x/delta-y you get 1 for the first again, but 1/2 for the second. Since math is somewhat arbitrary and we can make up whatever we want, I'm going to pick the first formula.

Sometimes these types of conventions lead to weird things. Math is full of some of these non-intuitive paradoxes. They're cool and interesting, but when presented out of context, students just lump them in with another crazy thing the teacher is asking us to believe. But what makes them cool and interesting is that they result from other conventions we have made because they made so much sense.

From where comes the hole!?!

For example: We go with the convention that right/up is positive and left/down is negative and we come up with the slope formula as above because it makes the most sense to do that. Well, what happens when you get a vertical line? Did it come from tilting your line more and more positive so that it is "the most positive you could ever get?" Or did it come from tilting your line the other way and making it more and more negative until it is "the most negative you could ever get?" From our basic understanding of positives and negatives these concepts are the two most opposite items we could think of with numbers, but here it stands visually as the same thing! At this point, we have two options. Accept the craziness of infinity and negative infinity and allow for weird things like that to happen OR redefine how we compute slope. Are we willing to come up with a more complicated way of computing a slope number that might resolve the infinity issue? Is there even a way to do that? It's an interesting conundrum.

There usually is a reason why math exists the way it does. Most of the math we teach in school has been around for hundreds, if not thousands, of years. It has resonated with people throughout cultures and time. But, at its root, the reason 1 + 1 = 2 is as deep as why the color red is more passionate than the color green.

Tuesday, March 1, 2011

The (Vampire) Tale of Calculus

The thing I love most about teaching calculus is that it seems to be the first course where it's easy to tell the story of the math. All the other courses leading into it seem to be a random collection of skills that are only loosely based on one another. For example, Algebra II (at least in our district) hops around from quadratics to exponentials to trigonometry, probability, systems of equations, and conics without much to tie them together.

Calculus flows very nicely, though.

We begin with our student adventurer moving to a new state and having to make new friends with abstract reasoning and rigorous work. Students find themselves walking along curves with limits only sometimes meeting their friends at an appointed spot and sometimes even walking towards a place that doesn't exist. The once familiar functions are treated as trivial, new functions are scarier and uglier than students thought possible making their much hated trig functions seem easy in comparison.

These infinitely (or infinitesmally) long walks along curves lead them to an otherworldly being: the derivative. At first the derivative is completely scary and unreasonable. "What does f(x+h) even mean?!" It's right there at the edge of their understanding and is beckoning them to come play, but has large wolverine-like teeth and is intimidating.

When students finally open up to the derivative's definition after walking the curves and even sometimes leaving the safety of the curves for tangents, they begin to see the true nature of derivatives. They discover the power rule. They are comfortable around the new creature, even when some of the scarier parts show up like the chain rule or even the quotient rule comes with a haunting melody to soothe its use.

Students finally let themselves fall in love with derivatives (which scarcely 2 months before none of them would've dreamed might exist) when it leaves them for a bit. Natural classroom forces tell students to leave the derivatives on the table while we go explore a new idea...

Their old friend, areas, come to visit. Brooding students want to return to derivatives as they begin to feel strange feelings towards Riemann sums, but we have left the derivative behind. Slightly different paths along accumulating functions where they step along thin rectangular blocks or trapezoidal strips lead them to a deeper appreciation of what area really means.

When students are finally beginning to forget about derivatives and are growing accustomed to loving the one they're with, the Fundamental Theorem of Calculus appears and reveals that these two supernatural beings are not only related, but at war with one another in an ancient struggle between infinities and zeros.

On the one hand, it is comforting to see the old derivative friend reappear. On the other, it is awkward to pair the old love with the new friend. The rest of the year is spent coming to terms with how the two relate and can become friends themselves.